The spin distribution of massive black holes ( MBHs ) contains rich information on the MBH growth history .
In this paper , we investigate the spin evolution of MBHs by assuming that each MBH experiences two-phase accretion , with an initial phase of coherent-accretion via either the standard thin disc or super-Eddington disc , followed by a chaotic-accretion phase composed of many episodes with different disc orientations .
If the chaotic-phase is significant to the growth of an MBH , the MBH spin quickly reaches the maximum value because of the initial coherent-accretion , then changes to a quasi-equilibrium state and fluctuates around a value mainly determined by the mean ratio of the disc to the MBH mass ( M _ { \bullet } ) in the chaotic-accretion episodes , and further declines due to late chaotic-accretion if M _ { \bullet } \gtrsim ( 1 - 3 ) \times 10 ^ { 8 } M _ { \odot } .
The turning point to this decline is determined by the equality of the disc warp radius and disc size .
By matching the currently available spin measurements with mock samples generated from the two-phase model ( s ) on the spin-mass plane , we find that MBHs must experience significant chaotic-accretion phase with many episodes and the mass accreted in each episode is roughly 1-2 percent of M _ { \bullet } or less .
MBHs with M _ { \bullet } \gtrsim 10 ^ { 8 } M _ { \odot } appear to have intermediate-to-high spins ( \sim 0.5 - 1 ) , while lighter MBHs have higher spins ( \gtrsim 0.8 ) .
The best matches also infer that ( 1 ) the radiative efficiencies ( \eta ) of those active MBHs appear to slightly decrease with M _ { \bullet } ; however , the correlation between \eta and M _ { \bullet } , if any , is weak ; ( 2 ) the mean radiative efficiency of active MBHs is \left < \eta \right > \sim 0.09 - 0.15 , consistent with the global constraints .